The Symmetric Unit and the Midline Theorem: A First-Principles Geometric Construction in which Classical ζ is the Limit of Finished Stations and the Riemann Hypothesis is the Statement that the Limit has all Non-Trivial Zeros on the Only Primary Self-Ratio Available
Abstract
The Symmetric Unit and the Midline Theorem A first-principles geometric construction in which the only primary self-ratio on a finished segment is 1/2. Classical ζ is identified later as a derived unit of comparison on that cut. In this order the Riemann Hypothesis is the statement that the limit has every non-trivial zero there, because no other self-ratio is available without privilege. This deposit timestamps a construction that did not begin as an attempt on the Riemann Hypothesis. It began from a question about privilege in a live model: when everything is moving, what is allowed to count as “now”? The question was pursued through elementary geometry, inversion, and a refusal to appoint a second privileged count. The master sequence is the only count allowed to go ahead. Everything else is dated against a station that has already finished. Abstract. The construction produces a rigid stock of measurements along a master sequence that cannot skip ahead. The central object is the Symmetric Unit. Its permanent cut is the unique primary, scale-invariant, ±-equal self-description of location on a segment: the midpoint ratio 1/2. The same unit carries a rigid similar triangle and a local circle. Later stations inherit that package at a larger radius. Trace comes first. Measurement second. Unit language third. Place is a coincidence of readings. Location is a place after a unit has been asked. A station is a finished prime on the master sequence — a bridge, not a zero. A live self-measure such as nπ/2 is a magnitude owned by the walk in the walk’s own ratio. A zeta zero is a later question: a derived unit assembled from independent processes and compared from a named HERE. Those meetings, when they occur, still sit on the only primary cut both sides already hold. Classical ζ is named only after the stock is built. The Midline Theorem is the statement that the limit has every non-trivial zero on the pull-back of the midpoint ratio. Forced identities are proved. Named constructions are labeled. No completed prime list and no external π are imported. Numerics demonstrate rigidity of the stock at finite depth. They are not a search for zero locations. This record contains. The paper, v2 (PDF and TeX). The script that rebuilds the appendix drawings from the stock. The live SU model, which walks that stock, writes the same CSVs, and exports a 3D coil whose height is the 1/2 axis. An extended drawing set read from the walk. Numerics that show rigidity, not zeros. This record does not contain. A claim that a gap midpoint is a zero. A claim that a live arc — 9π/2 at the apex of [7, 11], or any other self-measure — is a zeta zero. Version 1 remains the closed timestamp of the first writing. This version is the same construction with the order of language tightened, the drawings rebuilt from the stock, and the live machine included so the rigidity can be inspected.